Following [1]} a closed-form expression for the pdf can actually be written down in terms of certain special functions called Meijer \(G\) –functions. However, this expression is quite involved and requires extra work in practice for the derivation of bounds. In the following, we show that we can obtain simpler closed-form formulas for small values of \(M =1,2,3\) . They represent the most practical use cases in noncoherent communications. Moreover, we also provide a general recursive procedure to obtain the pdfs for higher values:
